Given the inequality;
[tex](x+1)(x+3)\ge0[/tex]We can begin by finding the signs of the factors;
[tex]\begin{gathered} \text{For (x+1);} \\ x+1=0\Rightarrow x=-1 \\ x+1<0\Rightarrow x<-1 \\ x+1>0\Rightarrow x>-1 \end{gathered}[/tex][tex]\begin{gathered} \text{For (x+3);} \\ x+3=0\Rightarrow x=-3 \\ x+3>0\Rightarrow x>-3 \\ x+3<0\Rightarrow x<-3 \end{gathered}[/tex]We can now identify the intervals that satisfy the required condition "greater than or equal to zero."
[tex]\begin{gathered} x<-3\text{ OR x}=-3 \\ x=-1\text{ OR x}>-1 \end{gathered}[/tex]This on the number line would now look like;
ANSWER:
[tex]\begin{gathered} x\le-3 \\ OR \\ x\ge-1 \end{gathered}[/tex]Expressing the number;
[tex]x\le-3[/tex]in set notation;
[tex]\mleft\lbrace x\in Z\mright|x\le-3\}[/tex]This means;
"x is a member of the set of integers such that x is less than or equla to negative 3."
The symbol that looks like an "E" means is a member of, the one that looks like a capital Z means set of integers, the slash means "such that...".
We would not use natural numbers because negatives do not occur naturally.
